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Gardner's deformation of the Krasil'shchik-Kersten system
The classical problem of construction of Gardner's deformations for
infinite-dimensional completely integrable systems of evolutionary partial
differential equations (PDE) amounts essentially to finding the recurrence
relations between the integrals of motion. Using the correspondence between the
zero-curvature representations and Gardner deformations for PDE, we construct a
Gardner's deformation for the Krasil'shchik-Kersten system. For this, we
introduce the new nonlocal variables in such a way that the rules to
differentiate them are consistent by virtue of the equations at hand and
second, the full system of Krasil'shchik-Kersten's equations and the new rules
contains the Korteweg-de Vries equation and classical Gardner's deformation for
it.
PACS: 02.30.Ik, 02.30,Jr, 02.40.-k, 11.30.-jComment: 7th International workshop "Group analysis of differential equations
and integrable systems" (15-19 June 2014, Larnaca, Cyprus), 19 page
On the (non)removability of spectral parameters in -graded zero-curvature representations and its applications
We generalise to the -graded set-up a practical method for
inspecting the (non)removability of parameters in zero-curvature
representations for partial differential equations (PDEs) under the action of
smooth families of gauge transformations. We illustrate the generation and
elimination of parameters in the flat structures over -graded
PDEs by analysing the link between deformation of zero-curvature
representations via infinitesimal gauge transformations and, on the other hand,
propagation of linear coverings over PDEs using the Fr\"olicher--Nijenhuis
bracket.Comment: 38 pages, accepted to Acta Appl. Mat
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